Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-02
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The complex exponential by its power series

Definition

Let ιR ⁣:N→R be the canonical-natural map of The canonical natural ι(n)=n⋅1F of a field, and let j ⁣:R→C be the real embedding j(x)=x+i0 from The complex numbers as R[x]/(x2+1), with the real embedding and imaginary unit i. For every n∈N, the factorial n! is nonzero by The factorial n! and the falling factorial nk‾, defined by recursion in N, so ιR(n!)>0 by Canonical naturals are positive and strictly increasing and consequently j(ιR(n!))≠0 in the complex field C=R[x]/(x2+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a−bi)/(a2+b2).

For z∈C, define exp⁡z:=∑n=0∞znj(ιR(n!)). whenever this complex series converges. Inside a complex expression we abbreviate the embedded denominator j(ιR(n!)) by n!, so the same definition may be written exp⁡z=∑zn/n! without identifying a natural number with a complex number. Powers and series are those of Integer powers in the complex field and Complex series, absolute convergence, complex power series, and radius of convergence. The convergence for every z∈C is discharged by The complex exponential series converges absolutely for every complex argument.

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Dependency tree · 2 levels

Sources